Homework Help:
Optical Phonons and Monatomic Model

Explain why there is no optical phonon in the dispersion curve for the one-dimensional monatomic chain of atoms.

3. The attempt at a solution

I am completely confused. I know that optical phonons get their name because when an array of 2 atoms of different charge oscillate they are eloctramagnetically active and can absorbe or emit infrared radiation.

So I could say optical phonons are impossible for monatomic chains because you would need two atoms of different charge to get phonons.

But, I think that is wrong. Because elsewhere I read that optical phonons are described by oscillations about a center of mass, while acoustic phonons are described by a translation of the center of mass. This must imply that monatomic chains cannot have oscillations about the atom pair's centers of mass. So now I'm thinking there must be some reason why two atoms of the same mass cannot oscillate about their center of mass....

But I have no idea what that reason would be! Just imagining two balls connected by a spring, it seems that they can oscillate about this center.

The photon has w/k=c, so w=ck. Essentially that's just a vertical line in an w-k graph, so any phonon branch that has w not equal to 0 as k-> 0 will be optically active.

That's just the problem. When I use my book's equation for the diatomic model, and make the masses the same, (Essentially reducing the diatomic model to monatomic... this is what a hint to a follow up problem says to do) I get two curves, one of which goes to 0 as k->0 but the other goes to a maximum at k->0. This is what I'm working with: